Sublinear Algorithms for Approximating String Compressibility

Sublinear Algorithms for Approximating String Compressibility

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Article ID: iaor20131269
Volume: 65
Issue: 3
Start Page Number: 685
End Page Number: 709
Publication Date: Mar 2013
Journal: Algorithmica
Authors: , , ,
Keywords: computers: data-structure, combinatorial analysis
Abstract:

We raise the question of approximating the compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run‐length encoding (RLE) and a variant of Lempel‐Ziv (LZ77), and present sublinear algorithms for approximating compressibility with respect to both schemes. We also give several lower bounds that show that our algorithms for both schemes cannot be improved significantly. Our investigation of LZ77 yields results whose interest goes beyond the initial questions we set out to study. In particular, we prove combinatorial structural lemmas that relate the compressibility of a string with respect to LZ77 to the number of distinct short substrings contained in it (its 𝓁th subword complexity , for small 𝓁). In addition, we show that approximating the compressibility with respect to LZ77 is related to approximating the support size of a distribution.

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